Problem: Simplify the following expression and state the condition under which the simplification is valid. $q = \dfrac{t^2 - 49}{t + 7}$
Solution: First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = t$ $ b = \sqrt{49} = 7$ So we can rewrite the expression as: $q = \dfrac{({t} + {7})({t} {-7})} {t + 7} $ We can divide the numerator and denominator by $(t + 7)$ on condition that $t \neq -7$ Therefore $q = t - 7; t \neq -7$